Achieving near-perfect light absorption in atomically thin transition metal dichalcogenides through band nesting

Near-perfect light absorbers (NPLAs), with absorbance, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{\mathcal{A}}}}}}}}$$\end{document}A, of at least 99%, have a wide range of applications ranging from energy and sensing devices to stealth technologies and secure communications. Previous work on NPLAs has mainly relied upon plasmonic structures or patterned metasurfaces, which require complex nanolithography, limiting their practical applications, particularly for large-area platforms. Here, we use the exceptional band nesting effect in TMDs, combined with a Salisbury screen geometry, to demonstrate NPLAs using only two or three uniform atomic layers of transition metal dichalcogenides (TMDs). The key innovation in our design, verified using theoretical calculations, is to stack monolayer TMDs in such a way as to minimize their interlayer coupling, thus preserving their strong band nesting properties. We experimentally demonstrate two feasible routes to controlling the interlayer coupling: twisted TMD bi-layers and TMD/buffer layer/TMD tri-layer heterostructures. Using these approaches, we demonstrate room-temperature values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{\mathcal{A}}}}}}}}$$\end{document}A=95% at λ=2.8 eV with theoretically predicted values as high as 99%. Moreover, the chemical variety of TMDs allows us to design NPLAs covering the entire visible range, paving the way for efficient atomically-thin optoelectronics.


S1. TRANSFER MATRIX METHOD
FIG. S1. Scattered fields at a dielectric interface. The boundary has a surface conductivity given by σ.
Consider scattering of an electromagnetic field at the dielectric interface shown in Fig. S1 S1 . For transverse magnetic fields, the magnetic field in each dielectric region is given by B (j) (r, t) = A j e ik j,z z + B j e −ik j,z z e i(qx−ωt)ŷ (S1) where k j,z , q are the z, x components of the wavevector related by k 2 j,z = ϵ j ω 2 /c 2 − q 2 . From Maxwell's equation, ∇ × B = µ 0 ϵ 0 ∂ t E, we find the electric field to be E (j) (r, t) = k j,z c 2 ωϵ j A j e ik j,z z − B j e −ik j,z z e i(qx−ωt)x .
The fields in the two dielectric regions are connected by the boundary conditions at z = 0 given as where η = ϵ 1 k 2,z /ϵ 2 k 1,z and ξ = σk 2,z /ωϵ 0 ϵ 2 . The matrix in the above equation is the transf ermatrix relating the field on either side of the dielectric interface The transmission and reflection amplitudes can be read off the transfer matrix as The transmitted and reflected intensities are then given by T = ϵ 1 k 2,z ϵ 2 k 1,z |t| 2 , R = |r| 2 .
In addition to a transfer matrix for a dielectric interface, we also need one for free space propagation. It is simple to see that this matrix is given by Any general layered structure can be completely described as a collection of dielectric interfaces and free space propagation. The total transfer matrix of a general layered structure will thus have the following form T tot = T 1→2 P(d 1 )T 2→3 P(d 2 ) · · · T (N −1)→N . Consider a three-layer structure where the first layer is air(ϵ 1 = 1), the second layer is a dielectric(ϵ 2 ) of thickness d, and the third layer is a metal(ϵ 3 ). For such a structure, we can expect resonant absorption when the reflected waves undergo destructive interference.
In terms of the dielectric thickness and free space wavelength, the resonance condition can be written as 2d where ϕ metal is the phase accumulated upon reflection off the metal and m is an integer. For a perfect conductor ϕ metal = π.
We now calculate the absorption for the setup described in the previous paragraph. The dielectric is set to be SiO 2 with ϵ 2 = 2.2 and thickness d = 500nm. The metal is first modeled using the Drude model with ω p = 9eV and γ = 10meV. We indeed find resonant absorption determined by the condition derived above (see Fig. S2). Similar results are seen when the metal is defined using experimental data for silver S2 with the addition of interband absorption from the metal.

S3. ANALYTIC CONDITION FOR PERFECT ABSORBER
Having shown numerically that near perfect absorption can be achieved, we would now like to find an analytic condition on the conductivity for perfect absorption. From the transfer matrix method, it may be shown that the analytic expression for reflection is Using the condition for destructive interference, i.e. e i2kd = −1, this expression is simplified If we assume that the metal is a perfect conductor ϵ 3 → ∞ In general, the conductivity is complex. So by writing σ = σ ′ + iσ ′′ the reflectivity is given by Hence we may conclude that for the condition kd = 2π(m+1/2), the absorption is maximized when σ ′ = ϵ 0 c √ ϵ 1 . The maximum of σ ′ coincides with a zero of σ ′′ due to the Kramers-Kronig relations which should give perfect absorption. If we assume σ ′′ → 0 and ϵ 1 = 1, the corresponding maximum absorption is simplified as and the absorbance approaches unity when σ ′ = ϵ 0 c.
1.5 2.0 2.5 3.5 ℏω(eV) The intrinsic value of ε 2 was normalized by ration between lattice constant of supercell and thickness of 2L MoS 2 . S3 Here, we used a 15×15×1 k-mesh and a total of 240 bands to obtain converged results.
The plane wave cutoff for the response function was chosen to be 266 eV.   To overcome the limitations of several micron-sized 2D materials with the commonly used scotch tape method, centimeter-sized monolayer TMD can be obtained using the Auassisted exfoliation method. S4,S5 At the first step, a highly polished (111) bare Si wafer was deposited with a 150 nm thick Au film as an ultra-flat template by e-beam evaporation (CHA industries, SEC 600). On top of the Au layer, polymethyl methacrylate (MicroChem, 950 PMMA C4) as protecting layer was spin-coated at a rate of 1000 r.p.m for 60 s. After baking at 120 • C for 2 min, the Au layer is peeling off from the Si substrate with blue tape. Then a freshly cleaved layered bulk MoS 2 was gently pressed on a freshly Au film to establish large-scale MoS 2 /Au contact, as shown in Fig. S5 (b). The monolayer MoS 2 is peeled off due to the interaction between Au and the sulfur atom of MoS 2 , which is stronger than the interlayer vdW interaction in bulk MoS 2 . The MoS 2 /Au film/PMMA/blue tape was transferred onto the desired substrate and then dipped in the acetone to dissolve the PMMA to peel off the blue tape. Finally, the top Au film was removed by aqueous KI/I 2 etchant (Sigma Aldrich, "Au etchant, standard") to release the MoS 2 .

S5. CHARACTERIZATION OF A LARGE SCALE MoS 2 (a) (b)
Intensity (a. u.) Photon energy (eV) To characterize the MoS 2 by Au film-assisted method, Fig. S6 (a) shows the Raman spectra with different thicknesses of MoS 2 . The finger print region in the Raman spectrum of 1L MoS 2 exhibits two main modes E 1 2g at 385 cm −1 and A 1g at 404 cm −1 with mode difference of 19 cm −1 . This clearly proves that it is monolayer MoS 2 , which is the same result as the commonly used scotch tape method. S6 It successfully reproduces the experimentally observed continuous red-shift E 1 2g and blue-shift A 1g with increased thickness, as shown by a dashed line. The frequency difference monotonically increases from 19 to 24 cm −1 with increasing thickness in the right panel of Fig. S6 (a), confirming the frequency difference as a reliable thickness indicator. The indirect-to-direct band gap crossover in 1L is a common property of all the semiconducting TMD investigated so far. Fig. S6 (b) shows room-temperature PL spectra at 532 nm excitation with different thicknesses of MoS 2 . The characteristic PL emission can be observed in 1L MoS 2 at 1.86 eV, corresponding to a direct transition at the band edge of κ point. As the number of layers increases, the PL emission peaks are red-shift at 1.83 (2L) and 1.79 eV (3L), respectively.  Optical contrast was calculated as (R-R 0 )/R 0 , and is directly proportional to the absorbance of the film, S7,S8 where R is the reflection spectra of the MoS 2 on the substrate, R 0 is the reflection spectra of the substrate, n is the refractive index of MoS 2 , and n 0 is the refractive index of the substrate. In order to support the hypothesis that the interlayer coupling contributes to the absorption with different twist angles, a control sample of twist 2L MoS 2 is assembled using standard dry-transfer techniques with a poly-propylene carbonate (PCC) stamp. As shown in AquaSave, a water-soluble conductive polymer, was spin-coated at a rate of 1,000 r.p.m for 1 min without baking. The PMMA/AquaSave was exposed to electron beams at 100 keV energy with an exposure dose of 650 µC/cm 2 using an electron-beam lithography system (Vistec, EBP5000+ To characterize these samples, Fig. S9   PMMA was removed with Acetone. Finally, MoS 2 by Au-assisted exfoliation method was picked up and transferred by dry transfer method. In Fig. S11(b) shows the Raman spectra of MoS 2 region (360-420 cm −1 ). In Fig. S11 (b), we observed the two Raman modes at approximately 383.7 and 402.1 cm −1 of 1L MoS 2 , corresponding to the E 1 2g and A 1g modes. The E 1 2g and A 1g modes after transferring graphene were almost identical compared to those of 1L MoS 2 . We confirmed the same trend from stacked top MoS 2 , regardless of the stacking angle. The Raman results show that the doping or strain of MoS 2 by graphene has a negligible effect on the shift of E 1 2g and A 1g modes. Fig. S11 (c) shows PL measurement of MoS 2 in heterostructure, which reveals significant quenching of over 50 % in intensity when compared to 1L MoS 2 . This is usually ascribed to charge transfer process. The considerable PL quenching can be attributed to the reduced recombination of e-h pairs through the heterojunction of graphene/MoS2. However, as shown in Fig. S12, it does not affect the light absorption of MoS 2 . Fig. S11(d) show Raman with c-c bonds of graphene: D, G, and 2D peak. The ratio of I 2D /I G is about 1.58, close to 2 of high-quality single-layer graphene.
To investigate the effect of graphene used as a buffer layer, we performed the optical contrast and absorbance with a comparison between 1L MoS 2 and MoS 2 /Gr heterostructure. and MoS 2 /Gr heterostructure(red line). Increasing the number of layers from 1L to 2L of MoS 2 , the energy of A exciton was slightly redshift due to the interlayer coupling. The B exciton, on the other hand, shows a slight blueshift. Interestingly, the C exciton peak also shifts and its dependence is even stronger than that of A and B exciton. In the case of MoS 2 /Gr heterostructure, all of the exciton (A, B, and C) position holds the same positions as those of 1L MoS 2 . Also, Fig. S12 (b) shows the absorbance of intrinsic (1L and 2L) MoS 2 and MoS 2 /Gr heterostructure. We confirmed that the absorbance of the MoS 2 /Gr heterostructure(red line) without exciton peak shift increases by 2.2 % compared to that of 1L MoS 2 (black line). In other words, it is clear evidence that graphene has no effect on the exciton absorbance of MoS 2 .

S9. MOLECULAR BEAM EPITAXY GROWTH
The WSe 2 /ZnSe/WSe 2 heterostructure growth was performed on a double side polished (DSP) sapphire (0001) wafers using molecular beam epitaxy (MBE). The sapphire was first degassed at 900 • C for 60 min and then ramped down to the growth temperature of 600 • C using a ramp rate of 30 • C/min. The growth process starts with the deposition of the bottom WSe 2 monolayer (1L) by co-depositing tungsten (W), evaporated using a multipocket e-beam evaporator, and elemental selenium (Se), evaporated using a cracker source.
A tungsten to selenium flux ratio of 1:200 was used. The W flux (∼ 5 × 10 −9 mbar) was obtained using an e-beam current of 200 mA and voltage of 6 kV while the Se flux (1 × 10 −6 mbar) was controlled by setting the temperature of the Se reservoir to 130 • C and its cracker tip to 1100 • C. The WSe 2 growth rate was 6 hours/1L. The 1L coverage was confirmed by the disappearance of the reflection high energy electron diffraction (RHEED) pattern of the sapphire substrate and the concomitant appearance of a new RHEED pattern corresponding to the WSe 2 . The buffer layer growth of 1 nm ZnSe was grown at 600 • C in the same chamber with the zinc (cell temperature: 320 • C, flux: 2×10 −6 mbar) and selenium (cell temperature: 190 • C, flux: 2 × 10 −7 mbar) co-evaporated using low-temperature Knudsen cells. A zinc to selenium flux ratio of 10:1 was used. The top 1L of WSe 2 was then grown at 600 • C on this ZnSe template using the same W e-beam evaporator and Se cracker source.
No post-growth annealing was performed to avoid any high temperature degradation of the bottom layers. To make the optical cavity, the sample was taken out of the MBE, moved to a plasma enhanced chemical vapor deposition (PECVD) chamber, and capped with 192 nm of SiO 2 deposited at 250 • C. Here, thickness of SiO 2 was chosen to be 192 nm, satisfying the critical coupling condition for the photon energy of 2.92 eV of B' exciton of WSe 2 . Lastly, 3 nm of titanium followed by 100 nm of silver was deposited using e-beam evaporation to be used as the back reflector.    and with Salisbury screen, respectively. As MoS 2 becomes thicker, the optical contrast continues to increase with gradual redshift, all of which are in good agreement with previous studies. S10,S11 The maximum optical contrast of twisted 2L MoS 2 and MoS 2 /Gr/MoS 2 heterostructure is almost the same as that of 3L MoS 2 with negligible redshift. Note that such a negligible redshift is practically useful in designing Salisbury screen since its optimum thickness of dielectric layer is the same as that of the monolayer case. We don't have experimental results on twisted 3L and 4L of MoS 2 . However, it is naturally deduced that twisted 3L MoS 2 should have higher optical contrast (or real part optical conductivity) than pristine 3L MoS 2 , and also should exhibit much weaker redshift.